On the 2-local property for operators on the space of some functions (Researches on isometries as preserver problems and related topics)

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(1)61 61. On the 2‐local property for operators on the space of some functions Takumi Uchiyama. Graduate School of Sience and Technology Niigata University This work was supported by the Research Institute for Mathematical Sciences,. a. Joint Usage/Research Center located in Kyoto University.. 1. Introduction. Šemrl introduced the 2‐locality [1] and defined 2‐local automorphisms and 2‐local derivations. For a given algebra A, T. :. any. a. (not necessarily linear nor multiplicative) map. Aarrow A. is said to be a 2‐local automorphism (resp. 2‐local derivation) if for. b\in A ,. there exists an automorphism (resp. derivation) T_{a,b} on Asuch that. a,. T(a)=T_{a,b}(a) and T(b)=T_{a,b}(b) . Šemrl proved that every 2‐local automorphism (resp. 2‐local derivation) of B(H) , the algebra of all bounded linear operators on an infinite dimensional separable Hilbert space. H,. is an automorphism (resp. deriva‐. tion). After that, Molnár extended the concept of 2‐locality to isometries [2] and studied 2‐local isometries of B(H) . If tive nor linear) map. T:Xarrow X. X. is a Banach space,. a. (not necessarily surjec‐. is called a 2‐local isometry if for any. x,. y\in X , there. exists a surjective complex‐linear isometry T_{x,y} on Xsuch that T(x)=T_{x,y}(x) and. T(y)=T_{x,y}(y) . Molnár showed that every 2‐local isometry is a complex‐linear isom‐ etry. Motivated by this result, Gyóry studied 2‐local isometries of the function space. C_{0}(X)[3] , where C_{0}(X) denotes the Banach algebra of all continuous complex‐valued functions vanishing at infinity on a locally compact Hausdorff space. X.. \acute{}. Gyó ry proved. that there exists a 2‐local isometry which is a non‐surjective complex‐linear isometry. on C_{0}(X) for some. X.. \acute{}. Gyó ry also proved that every 2‐local isometry of C_{0}(X) is a.

(2) 62 surjective complex‐linear isometry in the case of. X. is first countable a‐compact Haus‐. dorff space. We refer to other result [4, 5, 6, 7, 8]. By the Mazur‐Ulam theorem [9], every surjective isometry between normed spaces which preserves the origin is real‐. linear. So it would be interesting to consider real‐linear isometries. We motivated by a Kawamura, Koshimizu and Miura’s research of 2‐local isometries of C^{n}([0,1]). equiped with the. C ‐norm. (or. \Sigma ‐norm). [10] (notatations and the statement are in sec‐. tion 3) and studied 2‐local isometries of C^{1}([0,1]) as the 2‐locality of the group of all surjective “ real‐linear” isometries. We proved that every 2‐local isometries (as the. 2‐local property for the group of surjective real‐linear isometries) of C^{1}([0,1]) equiped with the. C ‐norm. (or. \Sigma ‐norm). is actually a surjective real‐linear isometry (Hosseini. studied in the case of different type norms [11]).. 2. 2‐local isometry. In this section, we prepare some definitions. We use the following notations for the given Banach space. X.. Iso_{\mathbb{C}}(X). :=. { T:Xarrow X| T:surjective complex‐linear isometry}. Iso_{\mathbb{R}}(X). :=. { T:Xarrow X| T:surjective real‐linear isometry}. By these notations, we can rewrite the definition of 2‐local isometry as follows.. Definition 2.1 (2‐local isometry(Molnár)). Let T:Xarrow X. X. be a Banach space. Then, a map. is a 2‐local isometry if the following holds.. \forall x, y\in X\exists T_{x,y}\in Iso_{\mathbb{C}}(X). s.t.. T(x)=T_{x,y}(x)\wedge T(y)=T_{x,y}(y). We want to consider 2‐local isometries as the 2‐local property for the group of all surjective “real‐linear” isometries in the main theorem. To emphasize it, we write. “ 2‐local Iso_{\mathbb{R}}(X) ” or “ 2‐local Iso_{\mathbb{C}}(X) ” instead of“ 2‐local isometry”’. 3. 2‐local isometries of the space of continuously differentiable functions. We need some preparations before writing the statement of a result of Kawamura, Koshimizu and Miura on 2‐local isometries of C^{n}([0,1]) equiped with the C ‐norm. (or. \Sigma ‐norm).. We denoted by C^{n}([0,1]) the space of all complex‐valued n ‐times con‐.

(3) 63 tinuously differentiable functions on the closed unit interval [0,1] . The C ‐norm and \Sigma ‐norm on. C^{n}([0,1]). are as follows.. \Vert f\Vert_{C}:=\sup_{t\in[0,1]}\sum_{k=0}^{n}\frac{|f^{(k)}| {k!}(f\in C^{n}([0, 1]) \Vert f\Vert_{\Sigma}:=\sum_{k=0}^{n}\frac{\Vert f^{(k)}(t)\Vert_{\infty} {k!} (f\in C^{n}([0,1]) ,. Both norms make C^{n}([0,1]) into a Banach algebra. Kawamura, Koshimizu and. Miura studied 2‐local isometries of this Banach algebra [10]. Theorem 3.1 (Kawamura, Koshimizu and Miura). Let. A. be a Banach algebra. (C^{n}([0,1]), \Vert\cdot\Vert_{C}) . Then every 2‐local Iso_{\mathbb{C}}(A)T is of the following form: T(f)=c[f\circ\pi]^{\varepsilon} (f\in C^{n}([0,1])) where. c\in \mathbb{T},. \pi\in. \{id, 1-id\}, \varepsilon\in\{\pm 1\}. and. ,. [f]^{\varepsilon} :=Re(f)+i\varepsilon Im(f) .. So every 2‐local Iso_{\mathbb{C} ((C^{n}([0,1]), \Vert . \Vert_{C})) map is in Iso_{\mathbb{C}}((C^{n}([0,1]), \Vert . \Vert_{C})) . They proved that this statement is also true for the. \Sigma ‐norm. in the case of. n=1 .. Moti‐. vated by this result, we studied 2‐local Iso_{\mathbb{R}}(C^{n}([0,1])) and proved that every 2‐local. Iso_{\mathbb{R}}(C^{1}([0,1])) map is in Iso_{\mathbb{R}}(C^{1}([0,1])) , where the norm is the 4. C ‐norm. or. \Sigma ‐norm.. Main theorem. We want to prove that every 2‐local Iso_{\mathbb{R}}(A) for. A. with the. C ‐norm. (or. \Sigma ‐norm). is. actually a surjective real‐linear isometry. Hereinafter, we denotes C^{1}([0,1]) equiped with the norm is C ‐norm or. \Sigma ‐norm. by A.. Theorem 4.1 (Main theorem). Every 2‐local Iso_{\mathbb{R}}(A) map S(f)=c[f\circ\pi]^{\varepsilon} (\forall f\in A) where. S. is of the form. ,. c\in \mathbb{T}, \varepsilon\in\{\pm 1\} and \pi\in\{1,1-id\}.. We applying two theorems in the proof of the main theorem. First one is a theorem. by Kawamura, Koshimizu and Miura [12]. This theorem gave the form of maps of. Iso_{\mathbb{R}}(C^{1}([0,1])). .. Theorem 4.2 (Kawamura, Koshimizu and Miura). Let. (C^{1}([0,1]), \Vert \Vert_{\Sigma}) . If. S. :. Barrow B. B. be (C^{n}([0,1]), \Vert \Vert_{C}) or. is a surjective real‐linear map, then there exists.

(4) 64 c\in \mathbb{T}, \pi\in\{1,1-id\} and \varepsilon\in\{\pm 1\} such that S(f)=c[f\circ\pi]^{\varepsilon} for every f\in B.. Second one is a theorem by Li, Peralta, Wang and Wang [13]. This theorem is an extension of a Kowalski‐Slodkowski’s theorem [14]. Theorem 4.3 (Li, Peralta, Wang and Wang). Let \mathb {C} ,. and let. (1). \triangle. \triangle. :. Barrow \mathbb{C}. B. be a unital Banach algebra on. be a mapping sutisfying the following properties:. : 1‐homogeneous (i.e. \triangle(\alpha x)=\alpha\triangle(x)). (2) \triangle(x)-\triangle(y)\in \mathbb{T}\sigma(x-y). (\forall x, y\in B). Then \triangle is linear, and there exists c\in \mathbb{T} such that c\triangle is multiplicative.. Now we prove the main theorem by applying above theorems. Proof of Theorem 4.1. Recall that norm or. \Sigma ‐norm.. A. is (C^{1}([0,1]), \Vert \Vert) , where the norm is the C‐. Let S be a 2‐local Iso_{\mathbb{R}}(A) map. For any pair f, g(\in A) , there exists. T_{f,g}\in Iso_{\mathbb{R}}(A) such that S(f)=T_{f,g}(f) and S(g)=T_{f,g}(g) . Applying Theorem 4.2, there exists c_{f,g}\in \mathbb{T}, \varepsilon_{f,g}\in\{\pm 1\} and \pi_{f,g}\in\{1,1-id\} such that T_{f,g} is of the. following form:. (1). T_{f,g}(h)=c_{f,g}[h\circ\pi_{f,g}]^{\varepsilon_{f,g}} (\forall h\in A) By this formula (1), we will show that the followings: S(\lambda f)=\lambda S(f). (\forall f\in A\forall\lambda\in \mathbb{C}) or S(\lambda f)=-\lambda S(f). (\forall f\in A\forall\lambda\in \mathbb{C}). (2). \sigma(S(f)-S(g))\in \mathbb{T}\sigma(f-g) (\forall f, g\in A). (3). \sigma(\overline{S}(f)-\overline{S}(g))\in \mathbb{T}\sigma(f-g) (\forall f, g\in A). (4). First, we show that (2) holds. Take any \lambda\in \mathbb{C}\backslash \{0\} and fix it. By the 2‐locality of. S. and Theorem 4.2, for every f(\in A) there exists T_{f,\lambda f}\in Iso_{\mathbb{R}}(A) which satisfies. S(\lambda f)=T_{f,\lambda f}(\lambda f)=c_{f}[(\lambda f)0\pi_{f}] ^{\varepsilon_{f}}=[\lambda]^{\varepsilon_{f}}T_{f,\lambda f}(f)=[\lambda] ^{\varepsilon_{f}}S(f). .. So S(\lambda f)=\lambda S(f) or S(\lambda f)=-\lambda S(f) holds. To show that (2), we suppose S(\lambda 1_{A})= \lambda S(1_{A}) and show that S(\lambda f)=\lambda S(f) holds for every f(\in A) . If there exists. A\backslash \{0\} such that S(\lambda F)=-\lambda S(F) , this. F. F\in. is not a real‐constant. We consider maps.

(5) 65 h_{s} :=sF+(1-s)1_{A}. (s\in[0,1]) . Because of. a real‐constant for all. s(\neq 0) .. F. is not a real‐constant, h_{s} is also not. Now we define. u= \sup\{t\in[0,1]|S(\lambda h_{s})=\lambda S(h_{s}) (0\leq\forall s\leq t)\}. By the definition of u , there exists two sequences \{f_{n}\},. \lambda S(f_{n}), S(\lambda g_{n})=-\lambda S(g_{n}). (\forall n\in \mathbb{N}). and. isometry, the following equations hold:. \{g_{n}\}\in A^{\mathbb{N} such that S(\lambda f_{n})=. n arrow\infty 1\dot{ \imath} mf_{n}=\lim_{narrow\infty}g_{n}=h_{u} .. Since S is an. \Vert S(\lambda f_{n})-S(\lambda g_{n})\Vert=\Vert\lambda f_{n}-\lambda g_{n} \Vert=|\lambda|\Vert f_{n}-g_{n}\Vertarrow 0 (narrow\infty) \Vert S(\lambda f_{n})+S(\lambda g_{n})\Vert=\Vert\lambda S(f_{n})-\lambda S(g_ {n})\Vert=|\lambda|\Vert f_{n}-g_{n}\Vertarrow 0. ,. (narrow\infty). .. Hence we have. \Vert 2S(\lambda f_{n})\Vert=\Vert S(\lambda f_{n})-S(\lambda g_{n})+S(\lambda f_{n})+S(\lambda g_{n})\Vert \leq\Vert S(\lambda f_{n})-S(\lambda g_{n})\Vert+\Vert S(\lambda f_{n})+ S(\lambda g_{n})\Vertarrow 0 (narrow\infty) so. 0= \lim_{narrow\infty}2S(\lambda f_{n})=2\lambda_{narrow\infty}1\dot{ \imath} mS(f_{n})=2\lambda h_{u}. holds.. Hence we have. ,. h_{u}=0 and. u\neq 0 , this is a contradiction. We have proved (2). Next, we prove that (3) holds.. Take any pair f, g\in A and fix them. By the 2‐locality of. T_{f,g}\in Iso_{\mathbb{R}}(A). S. and (1), There exists. such that. S(f)-S(g)=T_{f,g}(f)-T_{f,g}(g)=T_{f,g}(f-g)=c_{f,g}[(f-g)0\pi_{f,g}] ^{\varepsilon_{f,g}} Hence we have. \sigma(S(f)-S(g))=\sigma(c_{f,g}[(f-g)0\pi_{f,g}]^{\varepsilon_{f,g}}) =c_{f,g}[\sigma(f-g)]^{\varepsilon_{f,g}}\subset \mathbb{T}\sigma(f-g). .. Now we have proved (3) holds. Similarly, one can prove that (4) holds. Next, we define a map U:Aarrow A by. U:=\{ begin{ar ay}{l S,ifS(i1_{A})=iS(1_{A}) \overline{S},ifS(i1_{A})=-iS(1_{A}) \end{ar ay} Clearly. U. is also an isometry. We consider the composition mapping of. uation functional Aarrow \mathbb{C} .. \overline{U_{t}(1_{A})}U_{t}. \tau_{t}. : Aarrow \mathbb{C}, \tau_{t}(f) :=f(t). U. and eval‐. (t\in[0,1]) , and define U_{t}(=\tau_{t}\circ U) :. By (2),(3) and (4), each U_{t} satisfies the assumptions of Theorem 4.4. So is multiplicative. We prove that U(1_{A}) is a constant.. \sigma(U(1_{A}))\subset \mathbb{T} by.

(6) 66 (3) and (4), so \Vert U(1_{A})\Vert_{\infty}=1 . Hence we have 1=\Vert 1_{A}\Vert=\Vert U(1_{A})\Vert , and by the definition of the norm of. A,. we have \Vert U(1_{A})'\Vert_{\infty}=0 . This yields U(1_{A}) is a constant.. Now we proved that there exists c\in \mathbb{T} such that each cU_{t} is multiplicative. Since. the maximal ideal space M_{A} is homeomorphic to the under lying space [0,1] , we can define a map. \pi. : [0,1]arrow[0,1] satisfying cU_{t}=\tau_{\pi(t)} . Now. U(f)=c(f\circ\pi) (f\in A) Because. U. equals to S or \overline{S} , it suffices to prove that. the proof. By \pi=\overline{c}U(id), Because. \pi. is continuous,. \pi. \pi. because. U. V,. is id or. 1-id. to complete. must be surjective. Suppose not, we can take a point V. such that Vn{\rm Im}(\pi)=\emptyset . We can choose. and 0\neq\Vert f_{0}\Vert=\Vert U(f_{0})\Vert=\Vert c(f_{0}\circ\pi)\Vert=\Vert 0\Vert=0 holds. \Vert\pi'\Vert_{\infty}=1 . Hence, by the mean value theorem, \pi. .. \pi. is an isometry. This is a contradiction. So. contractivity of. is represented by. is differentiable and \Vert\pi\Vert=\Vert U(id)\Vert=\Vert id\Vert=2 holds.. s\in[0,1]\backslash {\rm Im}(\pi) and its neighborhood f_{0}\in A\backslash 0 as f_{0}=0 on. U. , one can prove that. \pi. is id or. \pi. \pi. is surjective. We also have. is contractive. By surjectivity and. 1-id.. \square. References. [1] P. Šemrl, Local automorphisms and derivations on B(H) , Proc. Amer. Math. Soc. 125 (1997) 2677‐2680. doi:10.1090/S0002‐9939‐97‐04073‐2 [2] L. Molnár, 2‐local isometries of some operator algebras, Proc. Edinb. Math. Soc. 45 (2002), no. 2, 349‐352. doi:10.1017/S0013091500000043 \acute{}. [3] M. Gyó ry, 2‐local isometries of C_{0}(X) , Acta Sci. Math. (Szeged) 67 (2001), no. 3‐4, 735‐746.. [4] O. Hatori, T. Miura, H. Oka and H. Takagi, 2‐local isometries and 2‐local au‐ tomorphisms on uniform algebras, Int. Math. Forum 2(50) (2007), 2491‐2502.. doi:10.12988/imf.2007.07219 [5] H. Al‐Halees and R. Fleming, tor valued function spaces,. J.. On 2‐local isometries on continuous vec‐ Math.. Anal.. Appl.. 354. (2009),. 70‐77. doi:10.1016/j. jmaa.2008.12.023 [6] A. Jiménez‐Vargas and M. Villegas‐Vallecillos,. 2‐local isometries on spaces of. Lipschitz functions, Canad. Math. Bull. 54 (2011), 680‐692. doi:10.4153/CMB‐ 2011‐025‐5.

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